Problem: Solve for $x$ : $ 4|x + 6| - 3 = 3|x + 6| + 10 $
Solution: Subtract $ {3|x + 6|} $ from both sides: $ \begin{eqnarray} 4|x + 6| - 3 &=& 3|x + 6| + 10 \\ \\ { - 3|x + 6|} && { - 3|x + 6|} \\ \\ 1|x + 6| - 3 &=& 10 \end{eqnarray} $ Add ${3}$ to both sides: $ \begin{eqnarray} 1|x + 6| - 3 &=& 10 \\ \\ { + 3} &=& { + 3} \\ \\ 1|x + 6| &=& 13 \end{eqnarray} $ Simplify: $ |x + 6| = 13$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 6 = -13 $ or $ x + 6 = 13 $ Solve for the solution where $x + 6$ is negative: $ x + 6 = -13 $ Subtract ${6}$ from both sides: $ \begin{eqnarray} x + 6 &=& -13 \\ \\ {- 6} && {- 6} \\ \\ x &=& -13 - 6 \end{eqnarray} $ $ x = -19 $ Then calculate the solution where $x + 6$ is positive: $ x + 6 = 13 $ Subtract ${6}$ from both sides: $ \begin{eqnarray} x + 6 &=& 13 \\ \\ {- 6} && {- 6} \\ \\ x &=& 13 - 6 \end{eqnarray} $ $ x = 7 $ Thus, the correct answer is $x = -19 $ or $x = 7 $.